M ay 2 00 9 Finite time extinction of the Kähler - Ricci flow 1
نویسنده
چکیده
We investigate the limiting behavior of the unnormalized Kähler-Ricci flow on a Kähler manifold with a polarized initial Kähler metric. We prove that the Kähler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kähler class is proportional to the first Chern class of the manifold. This proves a conjecture of Tian for the smooth solutions of the Kähler-Ricci flow.
منابع مشابه
ar X iv : 0 90 1 . 03 03 v 1 [ he p - th ] 5 J an 2 00 9 Flat BPS Domain Walls on 2 d Kähler - Ricci Soliton
In this paper we address several aspects of flat Bogomolnyi-PrasadSommerfeld (BPS) domain walls together with their Lorentz invariant vacua of 4d N = 1 supergravity coupled to a chiral multiplet. The scalar field spans a one-parameter family of 2d Kähler manifolds satisfying a Kähler-Ricci flow equation. We find that BPS equations and the scalar potential deform with respect to the real paramet...
متن کاملKähler-Ricci Flow with Degenerate Initial Class
In [2], the weak Kähler-Ricci flow was introduced for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is allowed to be no longer Kähler. The convergence as t → 0 is of great importance to study for this topic. 1 Motivation and Set-up Kähler-Ricci flow, the complex version of Ricci flow, has been under inten...
متن کاملun 2 00 6 On Kähler manifolds with positive orthogonal bisectional curvature
The famous Frankel conjecture asserts that any compact Kähler manifold with positive bisectional curvature must be biholomorphic to CP n. This conjecture was settled affirmatively in early 1980s by two groups of mathematicians independently: Siu-Yau[16] via differential geometry method and Morri [15] by algebraic method. There are many interesting papers following this celebrated work; in parti...
متن کامل2 7 N ov 2 00 6 MULTIPLIER IDEAL SHEAVES AND THE KÄHLER - RICCI FLOW
Multiplier ideal sheaves are constructed as obstructions to the convergence of the Kähler-Ricci flow on Fano manifolds, following earlier constructions of Kohn, Siu, and Nadel, and using the recent estimates of Kolodziej and Perelman.
متن کاملar X iv : 0 80 2 . 25 70 v 1 [ m at h . D G ] 1 9 Fe b 20 08 CANONICAL MEASURES AND KÄHLER - RICCI FLOW
We show that the Kähler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under biratio...
متن کامل